Sure, let's solve the problem step by step.
Given the marks obtained by the students, arranged in ascending order, including the expressions involving [tex]\( x \)[/tex]:
[tex]\[ 72, 74, 75, x-10, x+30, 85, 88, 94 \][/tex]
The median of a set of numbers is the middle value when the numbers are arranged in ascending order. Since there are 8 students, which is an even number, the median is the average of the 4th and 5th numbers in the list when arranged in ascending order.
1. The 4th number in the list is [tex]\( x - 10 \)[/tex].
2. The 5th number in the list is [tex]\( x + 30 \)[/tex].
We are given that the median mark is 82. Therefore, the average of the 4th and 5th numbers is equal to 82:
[tex]\[
\text{Median} = \frac{(x - 10) + (x + 30)}{2}
\][/tex]
Simplify the expression inside the fraction:
[tex]\[
\text{Median} = \frac{x - 10 + x + 30}{2}
\][/tex]
Combine like terms:
[tex]\[
\text{Median} = \frac{(2x + 20)}{2}
\][/tex]
Simplify by dividing each term by 2:
[tex]\[
\text{Median} = x + 10
\][/tex]
We are given that the median is 82:
[tex]\[
x + 10 = 82
\][/tex]
To find the value of [tex]\( x \)[/tex], solve this equation:
[tex]\[
x + 10 = 82
\][/tex]
Subtract 10 from both sides:
[tex]\[
x = 72
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is 72.
